Where and how can linear algebra be useful in practice.

Transcription

1 Where and how can linear algebra be useful in practice. Jiří Fiala All examples are simplified to demonstrate the use of tools of linear algebra. Real implementations are much more complex.

2 Systems of linear equations Weather forecast For weather forecasting the temperature, pressure, humidity and the speed and direction of airflow should be determined with sufficient accuracy based on the current weather reports from weather stations. Such situation could be described by a system of differential equations. Their exact solution is difficult and in some cases impossible.

3 Systems of linear equations For solving large systems of second order differential equations engineers use the finite element method. It it based on a division of the investigated space into small cells. On each cell the desired function is approximated by a linear combination of simple functions, e.g. piecewise linear. An approximation of a smooth function by a piecewise linear function. A decomposition of a piecewise linear function to a linear combination of simple functions. An example of a piecewise linear function on a triangulation in R 2

4 Systems of linear equations For modeling the airflow above the Central Europe, the space is divided by a three-dimensional grid of points (with horizontal pitch about 4,7 km). The total number of grid points and hence the number of the corresponding cells, is almost The resulting given system of differential equations is converted to a system of several hundreds millions of linear equations. Meteorologists at the ČHMÚ in Komořany need and can solve such a system. The weather forecast is updated every 6 hours according to the calculated model.

5 Basis change in a vector space Image compression in jpeg Jpeg (Joint photographic expert group) is a raster graphic format. Its principle is as follows: The image is first converted into the YCbCr color space. (YCbCr model was originally designed to transmit TV signals.) Each layer is cut into parts of size 8 8 pixels and each part is processed separately.

6 Basis change in a vector space Instead of encoding the intensity of 64 individual points the image of the entire piece is composed as a linear combination of 64 discrete harmonic functions above the 8 8 grid, which is the so-called discrete cosine Fourier transform Basis of individual points Basis of functions cos(ix) cos(jy) Small coefficients can be neglected, which leads the lossy compression. It has much better compression ratio than any lossless encoding, while the changes in the image could hardly be recognized by humans.

7 Eigenvalues Searching on Google Search for webpages by keywords is usually performed in two stages: 1. First, find all pages that contain those keywords. 2. Then sort this group by relevance. The calculation of the importance of pages in Google s search engine called the PageRank. was designed according to the Kendall Wei theory (±1950). Here the relevance of a page is directly proportional to the sum of relevances of pages that refer to it.

8 Eigenvalues Imagine that links between all web pages are encoded in a matrix M, where rows and columns correspond to single pages, s.t. { 1 if the i-th page refers on the j-th m i,j = 0 otherwise We get the so called adjacency matrix of the web. Relevances correspond to a nonnegative vector x, satisfying Mx = cx for a suitable positive constant c. Observe that x is the eigenvector of M corresponding to the eigenvalue c. It can be shown that with a minor modification of the matrix M, the eigenvalue c of the largest absolute value will be real and positive. Additionally, the eigenvector x corresponding to such eigenvalue has all its components positive.

9 Eigenvalues Some facts about the PageRank method. A draft of the method was designed by Larry Page and Sergey Brin. The project began in 1995, the first implementation has been tested in The vector x is updated continually, so that each component is recalculated once a month. The number of pages, i.e. the order the hypothetical matrix M in 2016 was of the magnitude of 4,75 billion. The matrix M is of course very sparse it has 7 links per page in average. In practice, the adjacency graph of the web is decomposed into smaller blocks, and even those are not represented by a matrix. The vector x is calculated by iterative approximate methods (25 80 iterations) with a fixed value of c = 0.85.

10 Integer linear programming Interference minimization in GSM networks A part of the wireless communication in GSM networks takes place between mobile phones and the co called BTS stations. A station is usually equipped with one or three antennas that serve the neighborhood of the station, called a cell. For the communication within the cell one or more frequencies are used, depending on the number of active GSM phones in the cell. On a single frequency, 6 8 phones can be served. Frequencies shall be allocated to transmitters so that no or only minimal interference appears. First, frequencies allocated to the transmitters of the same BTS should be treated. One shall also consider interference between the same or similar frequencies of close BTS s. The level of interference may depend on the surrounding terrain and other factors.

11 Integer linear programming A suitable plan for the frequency allocation can be obtained by integer linear programming methods. Binary variables x v f encode whether a transmitter v will be assigned frequency f. Further binary variables x v,v f,f which control the interference between two conflicting frequencies f and f at close transmitters v and v. The conditions of the ILP instance Ax = b guarantee that: each transmitter has available an appropriate number of frequencies variables x v,v f,f correctly determine if an interference appears. The objective function c T x is designed to minimize the number of positively evaluated variables x v,v f,f, i.e. to minimize the total interference.

12 Integer linear programming Realistic instances published on FAP web are of scale frequences for transmitters, i.e variables x v f conditions, i.e. variables x v,v f,f Frequency plans were compiled for some scenarios where the interference dropped by % w.r.t. to the plans used. Use one of these frequency plans led to the reduction of the overhead for the total handover of calls between cells by 20 %.

13 Further reading Finite element method, Jpeg, discrete cosine Fourier transform, Pagerank Wikipedia, Mathworld Google search engine structure Brin, Page: The anatomy of a large-scale hypertextual web search engine, Proc. WWW 7 (1998) Frequency allocation in GSM networks Eisenblätter: Assigning frequencies in GSM networks Oper. Research Proc. 2002, Springer (2003) Eisenblätter et al.: Frequency planning and ramifications of coloring, Disc. Math., Graph Th. 22(2002) 51 88